help with visualizing convex hull and polytope

I am currently taking a course in optimizations and have some problem with convex analysis.

The problem is I do not see the difference of a convex hull and polytope.

The definition of a convex hull is:

conv(V) := {a1*v1+....+ak*vk | a1,....,ak >= 0; a1+...+ak = 1}

so this gives a set. I seems i cannot see what this definition gives me (for example in 2D). for example: is the convex hull of 3 vertices in R^2 the set of points along the line of the convex hull (the convex hull itself) or is the convex hull of these 3 vertices all points enclosed by this hull?

Then how is it different from a polytope. Is a set a polytope if the set itself is the convex hull? meaning the set is convex.

I am currently taking a course in optimizations and have some problem with convex analysis.

The problem is I do not see the difference of a convex hull and polytope.

The definition of a convex hull is:

conv(V) := {a1*v1+....+ak*vk | a1,....,ak >= 0; a1+...+ak = 1}

so this gives a set. I seems i cannot see what this definition gives me (for example in 2D). for example: is the convex hull of 3 vertices in R^2 the set of points along the line of the convex hull (the convex hull itself) or is the convex hull of these 3 vertices all points enclosed by this hull?

Then how is it different from a polytope. Is a set a polytope if the set itself is the convex hull? meaning the set is convex.

Hope I am stating my problem clear enough.

I don't know much about "convex hull" but I do know that the Convex Hull of a "convex polytope" is itself. It refers to "non-convex" and groups of shapes.

It's something like the convex hull of a set of points (the vertices of the shape) is a smallest convex shape holding those points.